Tannaka-Krein theorem (in the narrow sense) is a particular Tannaka duality theorem for compact topological groups.
The original theorem is (in two independent and different formulations) stated and proved in
- Tadao Tannaka, Über den Dualitätssatz der nichtkommutativen topologischen Gruppen, Tohoku Math. J. 45 (1938), n. 1, 1–12 (project euclid has only Tohoku new series!)
- M.G. Krein, A principle of duality for bicompact groups and quadratic block algebras, Doklady AN SSSR 69 (1949), 725–728.
For a textbook treatment see
- E. Hewitt, K. A. Ross, Abstract harmonic analysis, 2 vols.